Munich Personal RePEc Archive
Computability of simple games: A
characterization and application to the
core
Kumabe, Masahiro and Mihara, H. Reiju
July 2006
Computability of simple games: A characterization
and application to the core
Masahiro Kumabe
Kanagawa Study Center, The University of the Air 2-31-1 Ooka, Minami-ku, Yokohama 232-0061, Japan
H. Reiju Mihara∗
Graduate School of Management, Kagawa University Takamatsu 760-8523, Japan
July, 2006
Abstract
It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coali-tions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives ex-amples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable simple games have a finite Nakamura number, im-plying that the number of alternatives that the players can deal with rationally is restricted.
Journal of Economic Literature Classifications: C71, D71, C69.
Keywords: Voting games, infinitely many players, recursion theory, Turing computability, computable manuals and contracts.
∗The corresponding author; the family name is “Mihara.” Phone: +81-87-832-1831.
1
Introduction
We investigate algorithmic computability of a particular class of coalitional games (cooperative games), called simple games (voting games). One can think of simple games as representing voting methods; alternatively, as rep-resenting “manuals” or “contracts.” We give a characterization of com-putable simple games and apply it to the theory of the core. For the latter application, we extend Nakamura’s theorem [34] regarding the core of simple games to the framework where not all subsets of players are deemed to be a coalition.
1.1 Computability analysis of social choice
Most of the paper (except the part on the theory of the core) can be viewed as a contribution to the foundations ofcomputability analysis of so-cial choice, which studies algorithmic properties of social decision-making. This literature includes Kelly [22], Lewis [27], Bartholdi et al. [11, 12], and Mihara [30, 31, 33], who study issues in social choice using recursion the-ory (the theory of computability and complexity, or study of algorithms). These works, which are mainly concerned with the complexity of rules or cooperative games in themselves, can be distinguished from the closely re-lated studies of the complexity of solutions for cooperative games, such as Deng and Papadimitriou [15] and Fang et al. [18]. (More generally, ap-plications of recursion theory to economic theory and game theory include Spear [42], Canning [13], Anderlini and Felli [1], Anderlini and Sabourian [2], Prasad [37], Richter and Wong [38, 39], and Evans and Thomas [17]. See also Lipman [28] and Rubinstein [40] for surveys of the literature on bounded rationality in these areas.)
1.2 Simple games with countably many players
Simple games have been central to the study of social choice (e.g., Banks [9], Austen-Smith and Banks [8], and Peleg [36]). Simple games on an algebra of coalitions of players assign either 0 or 1 to each coalition (member of the algebra). In the setting of players who face a yes/no question, a coalition intuitively describes those players who vote yes. A simple game is charac-terized by its winning coalitions—those assigned the value 1. (The other coalitions arelosing.) Winning coalitions are understood to be those coali-tions whose unanimous votes are decisive.
When there are onlyfinitelymany players, we can construct a finite table listing all winning coalitions. Computability is automatically satisfied, since such a table gives an algorithm for computing the game. The same argument does not hold when there are infinitely many players. Indeed, some simple games are noncomputable, since there are uncountably many simple games but only countably many computable ones (because each computable game is associated with an algorithm).
There are two typical approaches to introducing infinite population to a social choice model. In the “variable population” approach, players are po-tentially infinite, but each problem (or society) involves only finitely many players. Indeed, well-known schemes, such as simple majority rule, unanim-ity rule, and the Condorcet and the Borda rules, are all algorithms that apply to problems of any finite size. Kelly [22] adopts this approach, giving examples of noncomputable social choice rules. In the “fixed population” approach, which we adopt, each problem involves the whole set of infinitely many players. This approach dates back to Downs [16], who consider contin-uous voter distributions. The paper by Banks et al. [10] is a recent example of this approach to political theory.
in-teger, as desired. Our notion of computability (δ-computability) focuses on this class of coalitions—recursive coalitions—as well as the method (char-acteristic index) of describing them.
A fixed population of countably many players arises not only in vot-ing but in other contexts, such as a special class of multi-criterion decision making—depending on how we interpret a “player”:
Simulating future generations One may consider countably many play-ers (people) extending into the indefinite future.
Uncertainty One may consider finitely many persons facing countably many states of the world [30]: each player can be interpreted as a particular person in a particular state. The decision has to be made before a state is realized and identified. (This idea is formalized by Gomberg et al. [21], who introduce “n-period coalition space,” where
nis the number of persons.)
Team management Putting the right people (and equipment) in the right places is basic to team management.1 To ensure “due process” (which is sometimes called for), can a manager of a company write a “manual” (computable simple game2) elaborating the conditions that a team
must meet?
Fix a particular task such as operating an exclusive agency of the company.3 Ateam consists of members (people) and equipment. The manager’s job is to organize or give a licence to a team that satisfac-torily performs the task. Each member (or equipment) is described by attributes such as skills, position, availability at a particular time and place (in case of equipment such as a computer, the attributes may be the kind of operating system, the combination of software that may run at the same time, as well as hardware and network specifications). Each suchattribute can be thought of as a particular yes/no question, and there are countably many such questions.4
1
In line with much of cooperative game theory, we put aside the important problems of economics of organization, such as coordinating the activities of the team members by giving the right incentives.
2
Like Anderlini and Felli [1], who view contracts as algorithms, we view “manuals” as algorithms. They derive contract incompleteness through computability analysis.
3
Extension to finitely or countably many tasks is straightforward. Redefine ateamas consisting of members, equipment,andtasks. Then introduce a player for each task. Since a task can be regarded as a negative input, it will be more natural to assign 0 to those tasks undertaken and 1 to those not undertaken (think of the monotonicity condition).
4
Here, each player can be interpreted as a particular attribute of a particular member (or equipment).5 In other words, each coalition is identified with a 0-1 “matrix” of finitely many rows (each row specify-ing a member) and countably many columns (each column specifyspecify-ing a particular attribute).6
1.3 Overview of the results
Adopting the above notion of computability for simple games, Mihara [33] gives a sufficient condition and necessary conditions for computability. The sufficient condition [33, Proposition 5] is intuitively plausible: simple games with a finite carrier (such games are in effect finite, ignoring all except finitely many, fixed players’ votes) are computable. A necessary condition [33, Corollary 10] in the paper seems to exclude “nice” (in the voting context) infinite games: computable simple games have both finite winning coalitions and cofinite losing coalitions. He leaves open the questions (i) whether there exists a computable simple game that has no finite carrier and (ii) whether there exists a noncomputable simple game that has both finite winning coali-tions and cofinite losing coalicoali-tions. The first of these quescoali-tions is particularly important since if the answer were no, then only the games that are in effect finite would be computable, a rather uninteresting result. The answers to these questions (i) and (ii) are affirmative. We construct examples in Sec-tion 6 to show their existence. The construcSec-tion of these examples depends in essential ways on Proposition 4 (which gives a necessary condition for a simple game to be computable) or on the easier direction of Theorem 5 (which gives a sufficient condition). In contrast, the results in Mihara [33] are not useful enough for us to construct such examples.
Theorem 5 gives a necessary and sufficient condition for simple games to be computable. The condition roughly states that “finitely many, unnec-essarily fixed players matter.”
To explain the condition, let us introduce the notion of a “determining string.” Given a coalition S, its k-initial segment is the string of 0’s and 1’s of length k whose jth element (counting from zero) is 1 if j ∈ S and is 0 if j /∈ S. For example, if S = {0,2,4}, its 0-initial segment, 1-initial
example, such as elaborating the conditions for a certain medicine to take the desired effects and deciding whether a certain act is legal or not.
5
From the viewpoint of “due process,” it would be reasonable to define a simple game not for the set of (the names of) members but for the set of attributes. (This is particularly important where games cannot meet anonymity.) Considering characteristic games for the set of attributes (“skills”) can make it easy to express certain allocation problems and give solutions to them (see Yokoo et al. [45]).
6
segment, . . . , 8-initial segment, . . . are, respectively, the empty string, the string 1, the string 10, the string 101, the string 1010, the string 10101, the string 101010, the string 1010100, the string 10101000, . . . . We say that a (finite) stringτ iswinning determining if any coalitionGextendingτ (i.e.,τ
is an initial segment ofG) is winning. We define losing determining strings similarly.
The necessary and sufficient condition for computability according to Theorem 5 is the following: there are computably listable sets T0 of losing
determining strings and T1 of winning determining strings such that any
coalition has an initial segment in one of these sets. In the above example, the condition implies that at least one string from among the empty string, 1, 10, . . . , 10101000, . . . is in T0 or in T1—say, 1010 is in T1. Then any
coalition of which 0 and 2 are members but 1 or 3 is not, is winning. In this sense, one can determine whether a coalition is winning or losing by examining only finitely many players’ membership. In general, however,one cannot do so by picking finitely many players before a coalition is given.
Theorem 5 has an interesting implication for the nature of “manuals” or “contracts,” if we regard them as being composed of computable sim-ple games (e.g., the team management examsim-ple in Section 1.2). Consider how many “criteria” (players; e.g., member-attribute pairs) are needed for a “manual” to determine whether a given “situation” (coalition; e.g., team) is “acceptable” (winning; e.g., satisfactorily performs a given task). While increasingly complex situations may require increasingly many criteria, no situation (however complex) requires infinitely many criteria. The condi-tions (such as “infinitely many of the prime-numbered criteria must be met”) based on infinitely many criteria are ruled out.
The proof of Theorem 5 uses the recursion theorem. It involves much more intricate arguments of recursion theory than those in Mihara [33] giving only a partial characterization of the computable games.7
A natural characterization result might relate computability to well-known properties of simple games, such as monotonicity, properness, strong-ness, and nonweakness. Unfortunately, we are not likely to obtain such a result: as we clarify in a companion paper [26], computability is “unrelated to” the four properties just mentioned.
The earlier results [33] are easily obtained from Theorem 5. For exam-ple, if a computable game has a winning coalition, then, an initial segment of that coalition is winning determining, implying that (Proposition 9) the game has a finite winning coalition and a cofinite winning coalition. We give simple proofs to some of these results in Section 4. In particular, Propo-sition 12 strengthens the earlier result [33, Corollary 12] that computable games violate anonymity. (Detailed studies of anonymous rules based on
in-7
finite simple games include Mihara [29], Fey [19], and Gomberg et al. [21].)
1.4 Application to the theory of the core
Most cooperative game theorists are more interested in the properties of a
solution (or value) for games than in the properties of a game itself. In this sense, Section 5 deals with more interesting applications of Theorem 5. (Most of the section is of independent interest, and can be read without a knowledge of recursion theory.)
Theorem 16 is our main contribution to the study of acyclic preference aggregation rules in the spirit of Nakamura’s theorem [34] on the core of simple games. Banks [9], Truchon [43], and Andjiga and Mbih [3] are re-cent contributions to this literature. (Earlier papers on acyclic rules can be found in Truchon [43] and Austen-Smith and Banks [8].) Most works in this literature (including those just mentioned) consider finite sets of players. Nakamura [34] considers arbitrary (possibly infinite) sets of players and the algebra of all subsets of players. In contrast, we consider arbitrary sets of players andarbitrary algebras of coalitions.
Combining a simple game with a set of alternatives and a profile of in-dividual preferences, we define a simple game with (ordinal) preferences. Nakamura’s theorem [34] gives a necessary and sufficient condition for a simple game with preferences to have a nonempty core for all profiles: the number of alternatives is below a certain number, called theNakamura num-berof the simple game. We extend (Theorem 16) Nakamura’s theorem to the framework where simple games are defined on an arbitrary algebra of coali-tions (so that not all subsets of players are coalicoali-tions). It turns out that our proof for the generalized result is more elementary than Nakamura’s original proof; the latter is more complex than need be.
2
Framework
2.1 Simple games
LetN =N={0,1,2, . . .}be a countable set of (the names of) players. Any
recursive (algorithmically decidable) subset of N is called a (recursive) coalition.
Intuitively, a simple game describes in a crude manner the power distri-bution amongobservable (or describable) subsets of players. Since the cog-nitive ability of a human (or machine) is limited, it is not natural to assume that all subsets of players are observable when there are infinitely many play-ers. We therefore assume that only recursivesubsets are observable. This is a natural assumption in the present context, where algorithmic properties of simple games are investigated. According toChurch’s thesis [41, 35], the recursive coalitions are the sets of players for which there is an algorithm that can decide for the name of each player whether she is in the set.8 Note that the class REC of recursive coalitions forms aBoolean algebra; that is, it includesN and is closed under union, intersection, and comple-mentation. (We assume that observable coalitions are recursive, not just r.e. (recursively enumerable). Mihara [33, Remarks 1 and 16] gives three reasons: nonrecursive r.e. sets are observable in a very limited sense; the r.e. sets do not form a Boolean algebra; no satisfactory notion of computability can be defined if a simple game is defined on the domain of all r.e. sets.)
Formally, a(simple) gameis a collectionω ⊆REC of (recursive) coali-tions. We will be explicit when we require thatN ∈ω. The coalitions in ω
are said to bewinning. The coalitions not inω are said to belosing. One can regard a simple game as a function from REC to {0,1}, assigning the value 1 or 0 to each coalition, depending on whether it is winning or losing. We introduce from the theory of cooperative games a few basic notions of simple games [36, 44].9 A simple game ω is said to be monotonicif for
all coalitions S and T, the conditionsS ∈ω and T ⊇S imply T ∈ω. ω is
properif for all recursive coalitionsS,S∈ω impliesSc :=N\S /∈ω. ω is
strongif for all coalitions S,S /∈ω impliesSc ∈ω. ω is weakif ω=∅ or
the intersection ∩
ω =∩
S∈ωS of the winning coalitions is nonempty. The
members of∩
ω are called veto players; they are the players that belong
8
Soare [41] and Odifreddi [35] give a more precise definition ofrecursive setsas well as detailed discussion of recursion theory. Mihara’s papers [30, 31] contain short reviews of recursion theory.
9
to all winning coalitions. (The set ∩
ω of veto players may or may not be observable.) ω is dictatorial if there exists some i0 (called a dictator)
in N such that ω = {S ∈ REC :i0 ∈ S}. Note that a dictator is a veto
player, but a veto player is not necessarily a dictator.
We say that a simple game ω is finitely anonymous if for any finite permutation π : N → N (which permutes only finitely many players) and for any coalition S, we have S ∈ω ⇐⇒ π(S) ∈ω. In particular, finitely anonymous games treat any two coalitions with the same finite number of players equally. Finite anonymity is a notion much weaker than the version of anonymity that allows any (measurable) permutationπ :N →N. For example, free ultrafilters (nondictatorial ultrafilters) defined below are finitely anonymous.
A carrierof a simple gameω is a coalition S⊂N such that
T ∈ω ⇐⇒ S∩T ∈ω
for all coalitionsT. We observe that ifS is a carrier, then so is any coalition
S′⊇S.
Finally, we introduce a few notions from the theory of Boolean alge-bras [24]; they can be regarded as properties of simple games. A monotonic simple gameω satisfying N ∈ω and ∅∈/ω is called a prefilterif it has the finite intersection property: if ω′ ⊆ ω is finite, then ∩
ω′ 6= ∅. Intuitively,
a prefilter consists of “large” coalitions. A prefilter is freeif and only if it is nonweak (i.e., it has no veto players). A free prefilter does not contain any finite coalitions (Lemma 13). A prefilterω is afilterif it is closed with respect to finite intersection: ifS,S′ ∈ω, thenS∩S′ ∈ω. Theprincipal filter generated by S isω={T ∈REC :S⊆T}. It is a typical example of a filter that is not free; it has a carrier, namely,S. A filter is aprincipal filterif it is the principal filter generated by someS. A filterω is called an
ultrafilterif it is a strong simple game. Ifωis an ultrafilter, thenS∪S′∈ω
implies that S ∈ ω or S′ ∈ ω. An ultrafilter is free if and only if it is not
dictatorial.
2.2 Indicators for simple games
To define the notions of computability for simple games, we introduce be-low two indicators for them. In order to do that, we first represent each recursive coalition by a natural number: either by a characteristic index (∆0-index) or by an r.e. index (Σ1-index). Here, a number e is a
charac-teristic indexfor a coalitionSifϕe(the partial function computed by the
Turing program with code number e) is the characteristic function for S. Intuitively, a characteristic index for a coalition describes the coalition by a Turing program that can decide its membership. A number e is an r.e. indexfor a coalition S ifS =We :={x :ϕe(x) ↓ }, the domain of theeth
by a Turing program that will halt precisely when a member of the coalition is given as an input. A characteristic index gives more information about the coalition that it represents than an r.e. index does. The indicators then assign the value 0 or 1 to each number representing a coalition, depending on whether the coalition is winning or losing. When a number does not represent a recursive coalition, the value is undefined.
Given a simple game ω, its δ-indicatoris the partial functionδω on N
defined by
δω(e) =
1 ifeis a characteristic index for a recursive set in ω,
0 ifeis a characteristic index for a recursive set not in ω, ↑ ifeis not a characteristic index for any recursive set.
(1) Note thatδω is well-defined since each e∈Ncan be a characteristic index
(∆0-index) for at most one set. If we replace “a characteristic index” with
“an r.e. index” in (1), we obtain theσ-indicatorσω instead ofδω.
2.3 Computability notions
We now introduce the notions ofδ-computable simple games andσ-computable simple games.
We start by giving a scenario or intuition underlying the notion of δ -computability. (We can give a similar intuition for σ-computability.) A number (characteristic index) representing a coalition (equivalently, a Tur-ing program that can decide the membership of a coalition) is presented by an inquirer to the aggregator (planner), who will compute whether the coali-tion is winning or not. Though there is no effective (algorithmic) procedure to decide whether a number given by the inquirer is legitimate (i.e., rep-resents some recursive coalition), a human can often check manually (non-algorithmically) if such a number is a legitimate representation. We assume that the inquirer gives the aggregator only those indices that he has checked and proved its legitimacy. This assumption is justified if we assume that the aggregator always demands such proofs. The aggregator, however, cannot know a priori which indices will possibly be presented to her. (There are, of course, indices unlikely to be used by humans. But the aggregator cannot a priori rule out some of the indices.) So, the aggregator should be ready to compute whenever a legitimate representation is presented to her. This intuition justifies the following conditions of computability.
δ-computability δω has an extension to a partial recursive function.
σ-computability σω has an extension to a partial recursive function.
wheneverδω(e) is undefined) to be partial recursive [33, Appendix A]. Such a
condition cannot be satisfied, however, since the domain ofδω is not r.e. [30,
Lemma 2]. Sinceδ-indicators use more descriptive indices thanσ-indicators,
σ-computability impliesδ-computability [33, Lemma 2].
Remark 1. Multiple-choice or essay. One might argue that the scenario preceding the definitions of computability makes the aggregator’s task more difficult than need be. The difficulty comes from the fact that, like an essay exam, there is too much freedom on the side of the inquirer, the argument would go, in the sense that each recursive coalition has infinitely many indices and that the index presented may be an illegitimate one. An alternative notion of computability that deals with these problems might use a “multiple-choice format,” in which the aggregator gives possible indices that the inquirer can choose from. Unfortunately, such a “multiple-choice format” would not work as one might wish.
Indeed, we claim that there is no effective listinge0, e1, e2, . . . of
char-acteristic indices such that for each recursive coalitionS there is at least one
ei that represents the coalition (i.e., ei is a characteristic index for S). To
prove this claim, suppose there is such a listing and letS be the set defined by i∈S if and only if ϕei(i) = 0. Then sinceϕei(i) ↓ for anyi, we have S
recursive. On the other hand, the characteristic function forS is not equal to anyϕei. To see this, suppose that it is equal toϕei. Then, if i∈S, we
have ϕei(i) = 1, the definition of S then implies i /∈ S, a contradiction; if
i /∈S, we have a similar contradiction. The claim is thus proved.
Given this impossibility result, one might wish to relax the condition and allow some ei in the listing to fail to be a characteristic index. Adopting a
notion of computability based on such a listing is a halfway solution, fitting into neither the essay-exam scenario nor the multiple-choice alternative to it. k
The following results suggest thatσ-computability is too strong a notion of computability—due to lack of the descriptive power of r.e. indices.
Proposition 1 (Mihara [33, Proposition 3]) Suppose that N ∈ ω and
∅∈/ ω. Then the simple game ω is not σ-computable. In particular, proper simple games (for which N is winning) violate the condition.
Corollary 2 Suppose that N ∈ ω. If the simple game ω is σ-computable, then it is not proper. Furthermore, if it is monotonic, thenω = REC; that is, all coalitions are winning.
Turing program with the code number ewill never halt when the name of the dictator is given as an input.
We therefore focus on δ-computability in the remaining sections of the paper, discarding σ-computability.
3
A Characterization Result
Proposition 1 and its corollary indicate that σ-computable simple games have a rather uninteresting structure. We therefore investigateδ-computability in the rest of the paper.
3.1 Determining strings
The next lemma states that for any coalition S of a δ-computable simple game, there is a cutting number k such that any finite coalition G having the same k-players as S (that is, G and S are equal if players i ≥ k are ignored) is winning (losing) ifS is winning (losing). Note that ifkis such a cutting number, then so is anyk′ greater thank.
Notation. We identify a natural numberkwith the finite set{0,1,2, . . . , k−
1}, which is an initial segment of N. Given a coalition S ⊆ N, we write
S∩k to represent the coalition {i∈ S :i < k} consisting of the members of S whose name is less than k. We call S ∩k the k-initial segment of
S, and view it either as a subset of N or as the string S[k] of length k
of 0’s and 1’s (representing the restriction of its characteristic function to
{0,1,2, . . . , k−1}). Note that ifGis a coalition andG∩k=S∩k (that is,
Gand S are equal if players i≥k are ignored), the characteristic function of Gextends the k-initial segment (viewed as a string of 0’s and 1’s) ofS.
Lemma 3 Let ω be aδ-computable simple game. IfS∈ω, then there is an initial segmentk≥0ofNsuch that for anyfiniteG∈REC, ifG∩k=S∩k, thenG∈ω. Similarly, if S /∈ω, then there is an initial segment k≥0 of N
such that for any finite G∈REC, if G∩k=S∩k, then G /∈ω.
Proof. Let S ∈ω and assume for a contradiction that there is no such initial segment k. Then, for each initial segment k of N, there is a finite coalitionGk such that Gk∩k=S∩k and Gk ∈/ ω. Note that we can find
such Gk recursively (algorithmically) ink since it is finite.
Let K be a nonrecursive r.e. set such as{e:e∈We}. Since K is r.e.,
there is a recursive set R ⊆ N×N such that e ∈ K ⇔ ∃zR(e, z). Define
g(e, u) = µy ≤ u R(e, y) (i.e., the least y ≤ u such that R(e, y)) if such y
exists, andg(e, u) = 0 otherwise. Then g is recursive.
Using the Parameter Theorem, define a recursive functionf by
ϕf(e)(u) = 0 if¬∃z≤u R(e, z) and u6∈S,
ϕf(e)(u) = 1 if∃z≤u R(e, z) andu∈Gg(e,u), and
ϕf(e)(u) = 0 otherwise.
Now, on the one hand,e∈K implies thatf(e) is a characteristic index forGu′ ∈/ ωfor someu′. (Details: Givene∈K, letu′ =µy R(e, y), which is
well-defined since∃zR(e, z). Thenϕf(e)(u) = 1 iff (i)u < u′andu∈S[that
is, u < u′ and u ∈ G
u′] or (ii) u ≥ u′ and [since g(e, u) = u′ in this case]
u∈Gg(e,u)=Gu′. Thus ϕf(e)(u) = 1 iff u∈Gu′.) Henceδω(f(e)) = 0. On
the other hand, e /∈K implies thatf(e) is a characteristic index for S ∈ω. Hence δω(f(e)) = 1.
Sinceδω has an extension to a p.r. function (becauseωisδ-computable),
the last paragraph implies thatK is recursive. This is a contradiction. To prove the last half of the lemma, note that the set-theoretic difference ˆ
ω = REC−ω is also a δ-computable simple game. Let S /∈ ω. Then the first half applies to ˆω and S ∈ωˆ. SinceG∈ωˆ iff G /∈ω, the desired result follows.
In fact, the coalition G in Lemma 3 need not be finite. Before stat-ing an extension (Proposition 4) of Lemma 3, we introduce the notion of
determining strings:
Definition 1. Consider a simple game. A string τ (of 0’s and 1’s) of length k ≥ 0 is said to be determining if either any coalition G ∈ REC extendingτ (in the sense thatτ is an initial segment ofG, i.e.,G∩k=τ) is winning or any coalitionG∈REC extending τ is losing. A string τ is said to be determining for finite coalitions if either any finite coalition G
extending τ is winning or any finite coalition G extending τ is losing. A string is called nondeterminingif it is not determining.
Proposition 4 below states that for δ-computable simple games, (the characteristic function for) every coalition S has an initial segment S∩k
that is determining. (The number k−1 may be greater than the greatest element, if any, ofS):
Proposition 4 Suppose that a δ-computable simple game is given. (i) If a coalitionS is winning, then there is an initial segmentk≥0 ofNsuch that for any (finite or infinite) coalitionG, if G∩k=S∩k, then G is winning.
(ii) If S is losing, then there is an initial segment k ≥ 0 of N such that for any coalition G, if G∩k = S∩k, then G is losing. (iii) If S ∩k is an initial segment that is determining for finite coalitions, then S∩k is an initial segment that is determining.
Suppose S ∈ ω, where ω is a δ-computable simple game. Then by the first half of Lemma 3, there is k ≥ 0 such that (a) for any finite G′, if
G′∩k=S∩k, thenG′ ∈ω.
To obtain a contradiction, suppose that there isG /∈ω such that (b)G∩ k =S ∩k. By the last half of Lemma 3, there isk′ ≥0 such that (c) for
any finite G′, if G′∩k′ =G∩k′, then G′ ∈/ ω. Without loss of generality,
assumek′ ≥k.
Consider G′ = G∩k′, which is finite. Then, on the one hand, since
G′∩k′ =G∩k′, we getG′ ∈/ω by (c). On the other hand, since k′ ≥k, we
getG′∩k=G∩k=S∩k (the last equality by (b)). Then (a) implies that
G′∈ω. This is a contradiction.
3.2 Characterization of computable games
The next theorem characterizesδ-computable simple games in terms of sets of determining strings. Roughly speaking, finitely many players determine whether a coalition is winning or losing. Though we cannot tell in advance which finite set of players determines that, we can list such sets in an effective manner.
Note thatT0∪T1 in the theorem does not necessarily contain all
deter-mining strings. (The =⇒ direction can actually be strengthened: we can findrecursive, not just r.e., sets T0 and T1 satisfying the conditions. We do
not prove the strengthened result, since we will not use it.)
Theorem 5 A simple gameω isδ-computable if and only if there are an r.e. setT0 of losing determining strings and an r.e. setT1 of winning determining
strings such that (the characteristic function for) any coalition has an initial segment inT0 or in T1.
Remark 2. We can derive Theorem 5 from a result in Kreisel et al. [25] and Ce˘ıtin [14]. In this remark we largely follow the terminol-ogy of Odifreddi [35, pages 186–192 and 205–210], who gives a topolog-ical argument. In this remark only, a string refers to a finite sequence
σ=σ(0)σ(1)· · ·σ(k) of natural numbers (not necessarily 0 or 1).
Let PR be the class of partial recursive (unary) functions and R the class of recursive functions. An effective operation on R is a functional (function) F:R → Rsuch that for some partial recursive function ψ,
ϕe∈ R=⇒[ψ(e)↓and F(ϕe) =ϕψ(e)].
We introduce a topology into the set of partial (unary) functions by viewing it as a product space SN, with S = N∪ {↑}, ↑ being a
ifσ(x)↓, then f(x) =σ(x)} be the set of recursive functions that extendσ. These setsAσ are the basic open sets. Lettbe a recursive bijection between
the set N of natural numbers and the set of strings. We say a continuous functional F:PR → PR is effectively continuous on R ifF maps R to R
and for some recursive functionψ,
F−1(A
σ) ={f :f ∈Aν for someν ∈ {t(a) :a∈Wψ(t−1(σ))}} (2)
(requiring that the open sets F−1(A
σ) be obtained in a certain effective
way).
Kreisel et al. [25] and Ce˘ıtin [14] prove the theorem [35, Theorem II.4.6] stating thatthe effective operations on R are exactly the restrictions of the effectively continuous functionals onR.
In our context, suppose that ω is a δ-computable simple game. Let δ′
be a p.r. extension ofδω. Further let δ′′ be such thatδ′(x)↑ ⇔δ′′(x)↑ and
δ′(x) =i⇔δ′′(x) =e
i, where for eachi∈N,ei is an index of the constant
(recursive) function whose value is alwaysi. By the s-m-n theorem, define a recursive function g such that ϕg(e)(x) is 1, 0, or undefined, depending
on whether ϕe(x) is positive, zero, or undefined. In particular, if e is a
characteristic index, then g(e) is a characteristic index and ϕg(e) = ϕe.
Define F onR byF(ϕe) =ϕδ′′(g(e)). ThenF is an effective operation onR
(depending on whetherg(e) is a characteristic index for a winning coalition or a losing coalition, F(ϕe) =ϕe1 orF(ϕe) =ϕe0). By the theorem above,
F is effectively continuous on R so that for some recursive ψ, (2) holds. Denote byAithe setAσwhereσ=σ(0) =i. SinceF maps anyϕe∈ Rinto
constant functions ϕe1 ∈A1 and ϕe0 ∈A0, we haveF
−1({ϕ
ei}) =F
−1(A
i)
for i∈ {0,1}. We therefore have ϕe in F−1(A1) or in F−1(A0), depending
on whether e is a characteristic index for a winning coalition or a losing coalition. The =⇒ direction of Theorem 5 is obtained by letting Ti be the
r.e. set {t(a) :a∈Wψ(t−1
(i))}restricted to the 0-1 strings.
In this paper, we choose to give a different proof, which is perhaps more self-contained. The fact that the proof uses the recursion theorem should also be of some interest. k
Proof. (⇐=). We give an algorithm that can decide for each coalition whether it is winning or not: Given is a characteristic index e of a coali-tionS. Generate the elements ofT0 andT1; we can do that effectively since
these sets are r.e. Wait until an initial segment of S is generated. (Since a characteristic index is given, we can decide whether a string generated is an initial segment ofS.) If the initial segment is inT0, thenS is losing; if it is
inT1, then S is winning.
(=⇒). Supposeωisδ-computable. Letδ′ be a p.r. extension ofδ
ω; such
Overview. From Proposition 4, our goal is to effectively enumerate a determining initial segmentS∩k of each losing coalition S inT0 and that
of each winning coalition inT1.
We will define a certain recursive function y(e) in Step 1. In Step 2, we will first define the sets Ti, where i∈ {0,1}, as the collection of certain
strings (of 0’s and 1’s) of lengthk(e) (to be defined) for thosee∈Nsatisfying
δ′(y(e)) =i. In particular, T
0∪T1 includes, for each characteristic indexe,
thek(e)-initial segment of the recursive coalition indexed bye. We will then show thatT0 andT1 satisfy the conditions stated.
We use the followingnotation. We writeϕe,s(x) =y ifx,y,e < s and
y is the output ofϕe(x) in less thans steps of theeth Turing program [41,
p. 16]. We fix a Turing program forδ′ and denote byδ′
s(y) the computation
ofδ′(y) up to steps of the program.
Step 1. Defining a recursive function y(e).
We define a recursive functionf(e, y) in Step 1.1. In Step 1.2, we apply a variant of the Recursion Theorem to f(e, y) and obtain y(e).
Step 1.1. Defining a recursive functionf(e, y). Define an r.e. setQ0 ⊆N
by y ∈Q0 iff there exists s such that δs′(y) = 0 orδ′s(y) = 1. Define a p.r.
function
s0(y) =µs[δ′s(y)∈ {0,1}],
which converges fory∈Q0.
Fix a recursive set F of characteristic indices for finite sets such that each finite set has at least one characteristic index in F. (An example of
Fis the set consisting of the code numbers (G¨odel numbers) of the Turing programs of a particular form.) Fors∈N, let Fs=F∩s be the finite set
of numberse < s inF.
Define a set Q1 ⊆ N×N by (e, y) ∈ Q1 iff (i) y ∈ Q0 and (ii) there
exist s′ ≥ s
0 := s0(y) and e′ ∈ Fs′ such that (ii.a) δs′′(e′) = 1−δs′
0(y)
and that (ii.b) ϕe′,s′ is an extension of ϕe,s0−1. (Condition (ii.b), written
ϕe′,s′ ⊇ϕe,s0−1, means that if ϕe,s0−1(z) =u, thenϕe′,s′(z) =u.) Note that
if (e, y)∈Q1, then s0 =s0(y) is defined andδs′0(y) ∈ {0,1}. We can easily
check that Q1 is r.e. Given (e, y) ∈ Q1, let s1 be the least s′ ≥ s0 such
that conditions (ii.a) and (ii.b) hold for some e′ ∈F
s′. Let e0 be the least
e′ ∈F
s′ such that conditions (ii.a) and (ii.b) hold fors′ =s1. We can view
e0 as a p.r. function e0(e, y), which converges for (e, y)∈Q1.
Define the partial function ψ by
ψ(e, y, z) =
ϕe0(z) ify∈Q0 and (e, y)∈Q1,
ϕe,s0−1(z) ify∈Q0 and (e, y)∈/Q1,
ϕe(z) ify /∈Q0.
Proof. We show there is a sequence of p.r. functions ψs such that ψ =
∪
sψs. We then apply the Graph Theorem to concludeψ is p.r.
For each s ∈ N, define a recursive set Qs
0 ⊆ N by y ∈ Qs0 iff there
existss′ ≤ssuch that δ′
s′(y) = 0 or δ
′
s′(y) = 1. We havey ∈Q0 iff y∈Qs0
for somes. Note that if y∈Qs0 for somey, thens≥s0.
For each s ∈ N, define a recursive set Qs
1 ⊆ N×N by (e, y) ∈ Qs1 iff
(i)y ∈Qs
0 and (ii) there exist s′ such that s0 :=s0(y)≤s′ ≤sand e′ ∈Fs′
such that (ii.a)δ′s′(e
′
) = 1−δs′0(y) and that (ii.b) ϕe′,s′ is an extension of
ϕe,s0−1. (Conditions (ii.a) and (ii.b) are the same as those in the definition
ofQ1.) We have (e, y)∈Q1iff (e, y)∈Qs1for somes. Note that if (e, y)∈Qs1
for some (e, y), then s0≤s1 ≤s.
For eachs∈N, define the p.r. functionψs by
ψs(e, y, z) =
ϕe0,s(z) ify∈Q
s
0 and (e, y)∈Qs1,
ϕe,s0−1(z) ify ∈Q
s
0 and (e, y)∈/ Qs1,
ϕe,s(z) ify /∈Qs0.
We claim that ∪
sψs is a partial function (i.e.,
∪
sψs(e, y, z) does not take more than one value) and that ψ=∪
sψs:
• Suppose y /∈ Q0. Then y /∈ Q0s for any s. So, for all s, ψs(e, y, z) =
ϕe,s(z). Hence ∪sψs(e, y, z) =ϕe(z) =ψ(e, y, z) as desired.
• Suppose (y ∈ Q0 and) (e, y) ∈ Q1. Then s0, s1, and e0 are
de-fined and s1 ≥ s0. If s < s0, then since y /∈ Qs0 = ∅, we have
ψs(e, y, z) = ϕ
e,s(z). If s0 ≤ s < s1, then since y ∈ Qs0 and (e, y) ∈/
Qs
1 = ∅, we have ψs(e, y, z) = ϕe,s0−1(z). If s1 ≤ s, then since
y ∈ Qs
0 and (e, y) ∈ Qs1, we have ψs(e, y, z) = ϕe0,s(z). Hence
∪
sψs(e, y, z) =ϕe,s0−1(z)∪(
∪
s≥s1ϕe0,s(z)). The definition ofs1
im-plies that whens1≤s,ϕe,s0−1 ⊆ϕe0,s1 ⊆ϕe0,s. Thus
∪
sψs(e, y, z) =
∪
s≥s1ϕe0,s(z) =ϕe0(z) =ψ(e, y, z) as desired.
• Suppose y ∈ Q0 and (e, y) ∈/ Q1. Then s0 is defined. If s < s0,
then sincey /∈Qs
0 =∅, we have ψs(e, y, z) =ϕe,s(z). If s0 ≤s, then
sincey ∈Qs0 and (e, y)∈/ Qs1, we have ψs(e, y, z) =ϕe,s0−1(z). Hence
∪
sψs(e, y, z) =ϕe,s0−1(z) =ψ(e, y, z) as desired.
Define the partial function ˆψby ˆψ(s, e, y, z) =ψs(e, y, z). Then from the
construction of ψs, ˆψ is p.r. by Church’s Thesis. By the Graph Theorem,
the graph of ˆψ is r.e.
We claim that ψ = ∪
sψs is p.r. By the Graph Theorem it suffices to
show that its graph is r.e. We have (e, y, z, u) ∈ ψ ⇐⇒ ∃s (e, y, z, u) ∈ ψs⇐⇒ ∃s(s, e, y, z, u)∈ψˆ. Since the graph of ˆψ is r.e., it follows that the
Sinceψis p.r., there is a recursive functionf(e, y) such thatϕf(e,y)(z) =
ψ(e, y, z) by the Parameter Theorem.
Step 1.2. Applying the Recursion Theorem to obtain y(e). Since f(e, y) is recursive, by the Recursion Theorem with Parameters [41, p. 37] there is a recursive functiony(e) such thatϕy(e)=ϕf(e,y(e)). So, we haveϕy(e)(z) =
ψ(e, y(e), z).
We claim that the y = y(e) cannot meet the first case (y ∈ Q0 and
(e, y) ∈Q1) in the definition of ψ. Suppose y(e) ∈Q0 and (e, y(e))∈ Q1.
Sinceϕy(e)(z) =ψ(e, y(e), z), by the definition ofψwe have on the one hand
ϕy(e)=ϕe0. By (ii.a) of the definition of Q1 and by the definition ofe0, we
have on the other hand δ′(e0) = 1−δ′(y(e)) 6= δ′(y(e)). This contradicts
the fact thatδ′ extends the δ-indicatorδ
ω.
Therefore, we can express ϕy(e)(z) =ψ(e, y(e), z) as follows:
ϕy(e)(z) =
{
ϕe,s0−1(z) ify(e)∈Q0 ((e, y(e))∈/ Q1 implied),
ϕe(z) ify(e)∈/Q0. (3)
Step 2 Defining T0 and T1 and verifying the conditions.
Fori∈ {0,1}, letTibe the collection of all the stringsτ of lengthk(e) :=
s0 −1 (where s0 = s0(y(e))) that extendsϕe,s0−1 for all thosee such that
δ′(y(e)) =i. (Note that δ′(y(e))∈ {0,1} iffy(e)∈Q
0.) We showT0 and T1
satisfy the conditions.
Step 2.1. T0 and T1 are r.e. This is obvious since s0,δ′, and y are p.r.
(In other words, for each e, first find whether δ′(y(e))∈ {0,1}. If not, we
do not enumerate any segment in T0 or T1. If δ′(y(e)) = i ∈ {0,1}, then
we have corresponding strings whose length is effectively obtained. So we enumerate them inTi. This procedure ensures thatT0 and T1 are r.e.)
Step 2.2. T0 and T1 consist of determining strings. We show that T0
consists of losing determining strings. We can show that T1 consists of
winning determining strings in a similar way. Suppose δ′(y(e)) = 0. Then y(e) ∈ Q
0. Let s0 = s0(y(e)) and k(e) =
s0−1. Since (e, y(e))∈/ Q1 by (3), there is noe′ ∈F such thatδ′(e′) = 1−
δ′(y(e)) = 1 and thatϕ
e′is an extension ofϕe,s0−1. (Note thatδ
′(e′)∈ {0,1}
ife′ ∈F.) Hence any finite coalition that extendsϕ
e,s0−1 is losing.
Therefore, all stringsτ inT0(i.e., all the finite strings of lengthk(e) that
extend ϕe,s0−1 for some e such that δ
′(y(e)) = 0) are losing determining
for finite coalitions. By Proposition 4 (iii), all strings τ in T0 are losing
determining strings.
Step 2.3. Any coalition has an initial segment in T0 ∪T1. Let S be
a coalition, which is recursive. Pick a characteristic index e for S. We first show that δ′(y(e)) ∈ {0,1} (i.e., y(e) ∈ Q
(3), we have ϕy(e) = ϕe. So y(e) is a characteristic index for S. Hence
δ′(y(e))∈ {0,1}. That is, y(e)∈Q
0, which is a contradiction.
By the definitions of T0 andT1, sinceδ′(y(e))∈ {0,1}, all the strings of
lengthk(e) =s0−1 extendingϕe,s0−1 are inT0∪T1. In particular, since the
k(e)-initial segmentS∩k(e) of the characteristic functionϕe forS extends
ϕe,s0−1, the initial segmentS∩k(e) is inT0∪T1.
4
Applications: Finite Carriers, Finite Winning
Coalitions, Prefilters, and Nonanonymity
Theorem 5 is a powerful theorem. We can obtain as its corollaries some of the results in Mihara [33].
4.1 Finite carriers
The following proposition asserts that games that are essentially finite satisfy
δ-computability, as might be expected. We give here a proof that uses the characterization theorem.
Proposition 7 (Mihara [33, Proposition 5]) Suppose that a simple gameω
has a finite carrier. Then ω isδ-computable.
Proof. Suppose that ω has a finite carrier S. Letk= maxS+ 1 (we let
k= 0 if S =∅). Let
T1={τ :τ is a string of lengthkand τ ∈ω}
(whereτ ∈ωmeans that the set{i < k:τ(i) = 1}represented byτ, viewed
as a characteristic function, is inω) andT0 ={τ :τ is a string of lengthk and τ /∈ω}.
We verify the conditions of Theorem 5.
Since T0 and T1 are finite, they are r.e. Since T0 ∪T1 consists of all
strings of lengthk, any coalition has an initial segment in it.
We show thatT1 consists of winning determining strings. (We can show
that T0 consists of losing determining strings in a similar way.) Suppose
G∩k = τ ∈ T1. It suffices to show that G ∈ ω. By the definition of
T1, we have τ ∈ ω. This implies τ ∩S ∈ ω since S is a carrier. Since
τ∩S= (G∩k)∩S =G∩(k∩S) =G∩S, it follows that G∩S∈ω. Since
S is a carrier, we get G∈ω.
4.2 Finite winning coalitions
Note in Proposition 7 that if a game has a finite carrierSandN is winning, then there exists a finite winning coalition, namelyS=N∩S. When there does not exist a finite winning coalition, it is a corollary of the following negative result—itself a corollary of Theorem 5—-that the computability condition is violated.
Proposition 8 (Mihara [33, Proposition 6]) Suppose that a simple gameω
has an infinite winning coalitionS ∈ω such that for eachk∈N, itsk-initial segment S∩k is losing. Then ω is not δ-computable.
Proof. Suppose thatωisδ-computable. SinceS is winning, by Lemma 3 (or by Proposition 4 or by Theorem 5) there is some k ∈ N such that
G=S∩kis winning. This contradicts the assumption of the proposition.
Theorem 5 immediately gives the following extension of Corollary 7 of Mihara [33]. It gives a useful criterion for checking computability of simple games. Here, acofinite set is the complement of a finite set.
Proposition 9 Suppose that a δ-computable simple game has a winning coalition. Then, it has both finite winning coalitions and cofinite winning coalitions.
We also prove a result that is close to Proposition 8.
Proposition 10 (Mihara [33, Proposition 8]) Suppose that∅∈/ ω. Sup-pose that the simple game ω has an infinite coalition S ∈ ω such that for each k∈N, its difference S\k={s∈S :s≥k} from the initial segment is winning. Thenω is not δ-computable.
Proof. Supposeω is δ-computable. Since ∅∈/ ω, there is a losing deter-mining stringτ = 00· · ·0 of length kby Theorem 5. By assumption, S\k
is winning. But (S\k)∩k = τ and that τ is a losing determining string imply thatS\kis losing, which is a contradiction.
Again, the following proposition gives a useful criterion for checking computability of simple games.
4.3 Prefilters, filters, and ultrafilters
From the propositions in Section 4.2, examples of a noncomputable simple game are easy to come by.
Example 1. For anyqsatisfying 0< q≤ ∞, letωbe theq-complement ruledefined as follows: S ∈ω if and only if #(N \S)< q. For example, if
q= 1, then the q-complement rule is the unanimous game, consisting of N
alone. Ifq =∞, then the game consists of cofinite coalitions (the comple-ments of finite coalitions). Proposition 9 implies that q-complement rules are not δ-computable, since they have no finite winning coalitions. Any q -complement rule is a prefilter and it is a monotonic, proper, nonstrong, and anonymous simple game. Ifq >1, it is nonweak, but any finite intersection of winning coalitions is nonempty (i.e., has an infinite Nakamura number, to be defined in Section 5). Note that if 1< q <∞, then theq-complement rule is not a filter since it is not closed with respect to finite intersection. k
Example 1 gives examples of a prefilter that is not a filter. It also gives two examples of a filter that is not an ultrafilter: the unanimous game is a principal filter and the game consisting of all cofinite coalitions is a nonprincipal filter. Mihara [32] gives a constructive example of an ultrafilter. Some prefilters are computable, but that is true only if they have a veto player: according to Proposition 18 below, if a prefilter is δ-computable, then it is weak.
If ω is a filter, then it is δ-computable if and only if it is has a finite carrier [33, Corollary 11]. In particular, the principal filterω={T ∈REC :
S ⊆ T} generated by S has a carrier, namely, S. So, if S is finite, the principal filter is computable. If S is infinite, it is noncomputable, since it does not have a finite winning coalition. For example, the principal filter generated by S = 2N := {0,2,4, . . .} is a monotonic, proper, nonstrong, weak, and noncomputable simple game.
Ifωis a nonprincipal ultrafilter, it is notδ-computable by Proposition 9, since it has no finite winning coalitions (or no cofinite losing coalitions). It is a monotonic, proper, strong, and nonweak noncomputable simple game. In fact, an ultrafilter is δ-computable if and only if it is dictatorial [30, Lemma 4].
4.4 Nonanonymity
Proposition 12 Suppose that N ∈ ω and ∅ ∈/ ω. If the simple game ω is
δ-computable, then it is not finitely anonymous.
Proof. Let ω be a finitely anonymous δ-computable simple game such that N ∈ ω and ∅ ∈/ ω. Since N ∈ ω, there is an initial segment k :=
{0,1, . . . , k−1} = 11· · ·1 (string of 1’s of length k), by Lemma 3 (or by Proposition 4 or by Theorem 5). Since ∅∈/ ω, there is a losing determining stringτ = 00· · ·0 of lengthk′by Theorem 5. Then the concatenationτ∗k=
00· · ·011· · ·1 ofτ and k, viewed as a set, has the same number of elements as k. Since coalitions τ ∗k and k are finite and have the same number of elements, they should be treated equally by the finitely anonymousω. But
τ∗k is losing andk is winning.
5
The Number of Alternatives and the Core
In this section, we apply Theorem 5 to a social choice problem. We show (Corollary 17) that computability of a simple game entails a restriction on the number of alternatives that the set of players (with the coalition structure described by the simple game) can deal with rationally.
For that purpose, we define the notion of a simple game with (ordi-nal) preferences, a combination of a simple game and a set of alternatives and individual preferences. After defining the core for simple games with preferences, we extend (Theorem 16) Nakamura’s theorem [34] about the nonemptyness of the core: the core of a simple game with preferences is al-ways (i.e., for all profiles of preferences) nonempty if and only if the number of alternatives is finite and below a certain critical number, called the Naka-mura number of the simple game. We need to do this extension since what we call a “simple game” is not generally what is called a “simple game” in Nakamura [34].
We show (Corollary 15) that the Nakamura number of a nonweak simple game is finite if it is computable, though (Proposition 14) there is no upper bound for the set of the Nakamura numbers of such games. It follows from Theorem 16 that (Corollary 17) in order for a set of alternatives to always have a maximal element given a nonweak, computable game, the number of alternatives must be restricted. In contrast, some noncomputable (and nonweak) simple games do not have such a restriction (Proposition 18), and in fact have some nice properties. These results have implications for social choice theory; we suggest its connection with the study of Arrow’s Theorem [6].
5.1 Framework
Let N′ be an arbitrary nonempty set of players and B ⊆2N′
-simple game ω is a subcollection of B such that ∅ ∈/ ω. The elements of
ω are said to be winning, and the other elements inB are losing, as before. Our “simple game” is a B-simple game with N = N and B = REC, if it does not contain ∅. Nakamura’s “simple game” [34] is one with B = 2N′
. The properties (such as monotonicity and weakness, defined in Section 2.1) for simple games are redefined forB-simple games in an obvious way.
LetX be a (finite or infinite) set ofalternatives, with cardinal number #X ≥ 2. Let A be the set of (strict) preferences, i.e., acyclic (for any finite set {x1, x2, . . . , xm} ⊆X, if x1 ≻x2, . . . ,xm−1 ≻xm, then xm 6≻x1;
in particular,≻is asymmetric and irreflexive) binary relations ≻on X. (If
≻is acyclic, we can show that the relation º, defined by x ºy ⇔ y 6≻ x, is complete, i.e., reflexive and total.) A (B-measurable) profile is a list
p = (≻pi)i∈N′ ∈ AN ′
of individual preferences ≻pi such that {i ∈ N′ :
x≻pi y} ∈ B for all x,y∈X. Denote by AN′
B the set of all profiles.
A B-simple game with (ordinal) preferences is a list (ω, X,p) of a
B-simple gameω ⊆ B, a setX of alternatives, and a profile p= (≻pi)i∈N′ ∈
AN′
B . Given the B-simple game with preferences, we define the dominance
relation≻pω byx≻pωy if and only if there is a winning coalitionS ∈ω such
that x ≻pi y for all i∈ S. (In this definition, {i∈ N′ :x≻p
i y} need not
be winning since we do not assume ω is monotonic. Andjiga and Mbih [3] study Nakamura’s theorem, adopting the notion of dominance that requires the above coalition to be winning.) The core C(ω, X,p) of the B-simple game with preferences is the set of undominated alternatives:
C(ω, X,p) ={x∈X:6 ∃y∈X such thaty ≻pω x}.
A (preference) aggregation ruleis a map ≻:p7→ ≻p from profilesp
of preferences to binary relations (social preferences) ≻p on the set of
al-ternatives. For example, the mapping ≻ω from profiles p ∈ AN ′
B of acyclic
preferences to dominance relations≻pω is an aggregation rule. We typically
restrict individual and social preferences to those binary relations ≻on X
that are asymmetric (i.e., completeº) and either (i) acyclic or (ii) transitive (i.e., quasi-transitiveº) or (iii) negatively transitive (i.e., transitiveº). An aggregation rule is often referred to as asocial welfare function when indi-vidual preferences and social preferences are restricted to the asymmetric, negatively transitive relations.
5.2 Nakamura’s theorem and its consequences
Nakamura [34] gives a necessary condition for a 2N′
-simple game with pref-erences to have a nonempty core for any profilep, which is also sufficient if the setX of alternatives is finite. To state Nakamura’s theorem, we define the Nakamura numberν(ω) of a B-simple game ω to be the size of the smallest collection of winning coalitions having empty intersection
ν(ω) = min{#ω′:ω′ ⊆ω and ∩
if∩
ω =∅ (i.e., ω is nonweak); otherwise, set ν(ω) = #(2X) > #X. Note
that the Nakamura number is independent ofX andp.
The following useful lemma [34, Lemma 2.1] states that the Nakamura number of aB-simple game cannot exceed the size of a winning coalition by more than one.
Lemma 13 Let ω be a nonweak B-simple game. Then ν(ω) ≤ min{#S :
S∈ω}+ 1.
Proof. Choose a coalition S ∈ ω such that #S = min{#S : S ∈ ω}. Since ∩
ω = ∅, for each i ∈ S, there is some Si ∈ ω with i /∈ Si. So,
S∩(∩
i∈SSi) =∅. Therefore,ν(ω)≤#S+ 1.
It is easy to prove [34, Corollary 2.2] that the Nakamura number of a nonweakB-simple game is at most equal to the cardinal number #N of the set of players and that this maximum is attainable if B contains all finite coalitions. In fact, one can easily construct a computable, nonweak simple game with any given Nakamura number:
Proposition 14 For any integer k ≥ 2, there exists a δ-computable, non-weak simple game ω with Nakamura number ν(ω) =k.
Proof. Given an integerk≥2, let S={0,1, . . . , k−1}be a carrier and defineT ∈ω iff #(S∩T)≥k−1. Thenν(ω) =k.
Since computable, nonweak simple games have winning coalitions, it has finite winning coalitions by Proposition 9. An immediate corollary of Lemma 13 is the following:
Corollary 15 Let ω be a δ-computable, nonweak simple game. Then its Nakamura number ν(ω) is finite.
Nakamura [34] proves the following theorem forB= 2N′
:
Theorem 16 Let B be a Boolean algebra of sets ofN′. Suppose that ∅∈/ ω
and ω 6= ∅. Then the core C(ω, X,p) of a B-simple game (ω, X,p) with preferences is nonempty for all (measurable) profilesp∈ AN′
B if and only if
X is finite and #X < ν(ω).
the other necessary condition (disregarded by Nakamura) thatXis finite, we only need to considerfinite unions and intersections, and his proof actually works. Though accessible proofs are readily available in the literature (e.g., [8, Theorem 3.2]) for B = 2N′
and finite sets N′ of players, we choose
to give a proof here since most available proofs pay little attention to the
measurability condition (p∈ AN′
B ) for the profilesp that they construct. k
Proof. (⇐=). Suppose thatX is finite, #X < ν(ω), andC(ω, X,p) =∅
for some measurable profilep∈ AN′
B . Then follow the proof of Theorem 2.5
in Nakamura [34] to find a cycle with respect to ≻pω consisting of at most
#X alternatives.
(=⇒). SupposeC(ω, X,p)6=∅ for all p∈ AN′
B .
(i) To show that X is finite, suppose it is infinite. Then X contains a countable subset X′ ={x
1, x2, x3, . . .} ⊆X. Letp∈ AN
′
be a profile such that all playersi∈ N′ have an identical preference ≻pi (e.g., the transitive closure of itself) satisfyingxj+1 ≻pi xj for allj ∈ {1,2, . . .} and x1 ≻pi y for
all y ∈ X\X′. The measurability condition p ∈ AN′
B is satisfied since for
allx,y∈X, we have{i∈N′ :x≻p
i y}=N
′ or∅, both inB. Choose any
winning coalitionS∈ω, which exists by assumption. Then all players inS
have the same preference ≻pi, implyingxj+1≻pωxj for alljand x1 ≻pω y for
all y∈X\X′. It follows that C(ω, X,p) =∅; a contradiction.
(ii) To show that #X < ν(ω), supposer := #X ≥ν(ω). This excludes the possibility thatωis weak orν(ω) is infinite. We will construct a profilep
such that the dominance relation ≻pω has a cycle. By the definition of the
Nakamura number, there is a collection ω′ = {L
1, . . . , Lr} ⊆ ω such that
∩
ω′=∩r
k=1Lk=∅. DefineL0 =N′ and for allk∈ {1, . . . , r},
Dk= (L0∩L1∩ · · · ∩Lk−1)\Lk.
Then{D1, . . . , Dr}is a family of (possibly empty) pairwise disjoint coalitions
inB such that Lk ⊆Dkc :=N′ \Dk for allk and ∪rk=1Dk =N′ (i∈ N′ is
in the firstDk such that i /∈Lk).
Write X={x1, . . . , xr} and x0 =xr. Fix the cycle
≻={(xk, xk−1) :k∈ {1, . . . , r}}.
Define p ∈ AN′
as follows: for each k, all players i in Dk have the same
(acyclic) preference ≻pi=≻ \{(xk, xk−1)}. Then for all (x, y) ∈ ≻/ , we have
{i∈N′ :x≻p
i y}=∅ ∈ B. On the other hand, for all (x, y) = (xk, xk−1)∈
≻, we have {i∈N′ :x≻p
i y}=Dck∈ B and Lk⊆Dkc. Therefore, p∈ AN ′
B
and ≻pω=≻, a cycle. It follows that C(ω, X,p) =∅; a contradiction.
It follows from Theorem 16 that if a B-simple game ω is weak (and satisfies ∅ ∈/ ω and ω 6= ∅), then the core C(ω, X,p) is nonempty for all profilesp∈ AN′
ω is nonweak. Combined with Corollary 15, Theorem 16 has a consequence for nonweak, computable simple games:
Corollary 17 Let ω be a δ-computable, nonweak simple game satisfying
∅ ∈/ ω and ω 6= ∅. Then there exists a finite number ν (the Nakamura number ν(ω)) such that the core C(ω, X,p) is nonempty for all profilesp∈ AN
REC if and only if #X < ν.
If we drop the computability condition, the above conclusion no longer holds. An example of ω that has no such restriction on the size of the set X of alternatives is a nonweak prefilter (e.g., the q-complement rule of Example 1, forq >1), which has an infinite Nakamura number.
In fact, we can say more, if we shift our attention from the core—the set of undominated alternatives with respect to the dominance relation≻pω—to
the dominance relation itself.
Proposition 18 Let ω be a nonweak simple game satisfying∅∈/ ω andω6=
∅. (i) ω cannot be a δ-computable prefilter. (ii) If ω is δ-computable; then
ν(ω)is finite, and ≻pω is acyclic for allp∈ ANREC if and only if #X < ν(ω).
(iii)If ω is a prefilter, then≻pω is acyclic for allp∈ ANREC, regardless of the cardinal number#X of X.
Proof. (i) If ω is a nonweak prefilter, then it has an infinite Nakamura number. But nonweak computable games have finite Nakamura number by Corollary 15.
(ii) and (iii) are obvious from the following corollary [34, Theorem 3.1] of Theorem 16: ≻pω is acyclic for all p∈ AN
′
B if and only if #X
′< ν(ω) for
all finiteX′ ⊆X. (This corollary can also be obtained from the well-known fact that≻pωis acyclic if and only if the setC(ω, X′,p) of maximal elements
with respect to≻pω is nonempty for all finite subsetsX′ ofX.)
We can strengthen the acyclicity of the dominance relation ≻pω in
state-ment (iii) of Proposition 18 by replacing the statestate-ment with one of the following: (iv) if ω is a filter, then ≻pω is transitive for all p such that all
individuals have transitive preferences≻pi; (v) ifω is anultrafilter, then≻pω
is asymmetric and negatively transitive for all p such that all individuals have asymmetric, negatively transitive preferences≻pi. In fact, statements (iii), (iv), and (v) each gives an aggregation rule ≻ω:p7→ ≻p that satisfies
In an attempt to escape from Arrow’s impossibility, many authors have investigated the consequences of relaxing the rationality requirement (neg-ative transitivity of≻pω) for social preferences. In view of the close
connec-tion between the raconnec-tionality properties of an aggregaconnec-tion rule and preflters [8, Theorems 2.6 and 2.7] (also [23, 4, 5]), Proposition 18 has a significant implication for this investigation.
6
Examples
Propositions 7, 9, and 11 show that the class of computable games (i) in-cludes the class of games that have finite carriers and (ii) is included in the class of games that have both finite winning coalitions and cofinite los-ing coalitions. In this section, we construct examples showlos-ing that these inclusions are strict.
We can find such examples without sacrificing the desirable properties of simple games. We pursue this task thoroughly in a companion paper [26]. Thenoncomputable simple game example in Section 6.1 that has both finite winning coalitions and cofinite losing coalitions is a sample of that work. It is monotonic, proper, strong, and nonweak. An example of a computable
simple game that is monotonic, proper, strong, nonweak, and has no finite carrier is given in that work [26].
6.1 A noncomputable game with finite winning coalitions
We exhibit here a noncomputable simple game that is monotonic, proper, strong, nonweak, and have both finite winning coalitions and cofinite los-ing coalitions. It shows in particular that the class of computable games is strictly smaller than the class of games that have both finite winning coalitions and cofinite losing coalitions. In this respect, the game is unlike nonweak prefilters (such as the q-complement rules in Example 1); those examples do not have any finite winning coalitions. Furthermore, unlike nonprincipal ultrafilters—which are also monotonic, proper, strong, and nonweak noncomputable simple games—the game is nonweak in a stronger sense: it violates the finite intersection property.
LetA=N\ {0}={1,2,3, . . .}. We define the simple gameωas follows: Any coalition except Ac ={0} extending the string 1 of length 1 (i.e., any
coalition containing 0) is winning; any coalition except A extending the string 0 is losing; A is winning and Ac is losing. In other words, for all
S∈REC,
S ∈ω ⇐⇒ [S=A or (0∈S &S 6=Ac)].
of the extensive form of an infinitely repeated game played by you, with the stage game consisting of two moves 0 and 1. If you choose 1 in the first stage, you will win unless you keep choosing 0 indefinitely thereafter; if you choose 0 in the first stage, you will lose unless you keep choosing 1 indefinitely thereafter. Now, you “represent” a certain coalition and play 1 in stage i ifi is in the coalition; you play 0 in that stage otherwise. Then the coalition that you represent is winning if you win; it is losing if you lose.
k
Lemma 19 ω is notδ-computable.
The following proof demonstrates the power of Theorem 5, although its full force is not used (Proposition 4 suffices). Proposition 8, which appeared earlier in Mihara [33], does not have this power.
Proof. If ω is δ-computable, then by Theorem 5 (or by Proposition 4),
A has an initial segment A∩k that is a winning determining string. But
A∩k itself is not winning (though it extends the string trivially).
Lemma 20 ω has both finite winning coalitions and cofinite losing coali-tions.
Proof. For instance,{0,1}is finite and winning. N\{0,1}={2,3,4, . . .}
is cofinite and losing.
Lemma 21 ω is monotonic.
Proof. Suppose S∈ω andS (T. There are two possibilities. IfS =A, then T = N, and we have N ∈ ω by the definition of ω. Otherwise, S
contains 0 and some other numberi. The same is true ofT, implying that
T ∈ω.
Lemma 22 ω is proper and strong.
Proof. It suffices to show that Sc ∈ω ⇐⇒ S /∈ω. From the definition
ofω, we have
S /∈ω ⇐⇒ S6=A& (0∈/ S orS=Ac)
⇐⇒ Sc 6=Ac & (0∈Sc orSc =A)
⇐⇒ (0∈Sc &Sc 6=Ac) or Sc =A
